Skip to main content

First passage times of a jump diffusion process

  • S. G. Kou (a1) and Hui Wang (a2)

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

Corresponding author
Postal address: Department of IEOR, Columbia University, New York, NY 10027, USA. Email address:
∗∗ Postal address: Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA.
Hide All
[1] Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 588.
[2] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.
[3] Asmussen, S., Glynn, P. and Pitman, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Prob. 5, 875896.
[4] Bateman, H. (1953). Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York.
[5] Bateman, H. (1954). Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York.
[6] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
[7] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705766.
[8] Boyarchenko, S. and Levendorskii, S. (2002). Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Prob. 12, 12611298.
[9] Brémaud, P., (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.
[10] Duffie, D. (1995). Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press.
[11] Gerber, H. and Landry, B. (1998). On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance Math. Econom. 22, 263276.
[12] Glasserman, P. and Kou, S. G. (2003). The term structure of simple forward rates with jump risk. To appear in Math. Finance.
[13] Hull, J. C. (1999). Options, Futures, and Other Derivative Securities, 4th edn. Prentice-Hall, Englewood Cliffs, NJ.
[14] Ikeda, N. and Watanabe, S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2, 7995.
[15] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
[16] Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.
[17] Karlin, S. and Taylor, H. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.
[18] Kou, S. G. (2002). A jump diffusion model for option pricing. Manag. Sci. 48, 10861101.
[19] Kou, S. G. and Wang, H. (2001). Option pricing under a double exponential jump diffusion model. Preprint, Columbia University and Brown University.
[20] Merton, R. C. (1976). Option pricing when the underlying stock returns are discontinuous. J. Financial Econom. 3, 115144.
[21] Pecherskii, E. A. and Rogozin, B. A. (1969). On joint distributions of random variables associated with fluctuations of a process with independent increment. Theory Prob. Appl. 15, 410423.
[22] Pitman, J. W. (1981). Lévy system and path decompositions. In Seminar on Stochastic Processes (Evanston, IL, 1981), Birkhäuser, Boston, MA, pp. 79110.
[23] Protter, P. (1990). Stochastic Integration and Differential Equations. A New Approach (Appl. Math. 21). Springer, Berlin.
[24] Rogers, L. C. G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37, 11731180.
[25] Rogozin, B. A. (1966). On the distribution of functionals related to boundary problems for processes with independent increments. Theory Prob. Appl. 11, 580591.
[26] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
[27] Siegmund, D. (1985). Sequential Analysis. Springer, New York.
[28] Weil, M. (1971). Conditionnement par rapport au passé strict. In Séminaire de Probabilités V, Université de Strasbourg, année universitaire 1969–1970 (Lecture Notes Math. 191). Springer, Berlin, pp. 362372.
[29] Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis (CBMS-NSF Regional Conf. Ser. Appl. Math. 39). SIAM, Philadelphia, PA.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed